LP decay for general hyperbolic-parabolic systems of balance laws

Yanni Zeng

doi:10.3934/dcds.2018018

Article Contents

2018, Volume 38, Issue 1: 363-396. Doi: 10.3934/dcds.2018018

This issue Previous Article Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one Next Article Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane

L^P decay for general hyperbolic-parabolic systems of balance laws

Yanni Zeng

Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA

Received: April 30, 2017

Revised: July 31, 2017

Published: October 2017

This work was partially supported by a grant from the Simons Foundation (#244905 to Yanni Zeng)

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Abstract

We study time asymptotic decay of solutions for a general system of hyperbolic-parabolic balance laws in multi space dimensions. The system has physical viscosity matrices and a lower order term for relaxation, damping or chemical reaction. The viscosity matrices and the Jacobian matrix of the lower order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper we obtain optimal $L^p$ decay rates for $p≥2$. Our result is general and applies to physical models such as gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively.

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Mathematics Subject Classification: Primary:35B40, 35M31;Secondary:35Q35.

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